Exterior Angle Theorem
The exterior angle theorem states that when a triangle's side is extended, the resultant exterior angle formed is equal to the sum of the measures of the two opposite interior angles of the triangle. The theorem can be used to find the measure of an unknown angle in a triangle. To apply the theorem, we first need to identify the exterior angle and then the associated two remote interior angles of the triangle.
| 1. | What is Exterior Angle Theorem? |
| 2. | Proof of Exterior Angle Theorem |
| 3. | Exterior Angle Inequality Theorem |
| 4. | FAQs on Exterior Angle Theorem |
What is Exterior Angle Theorem?
The exterior angle theorem states that the measure of an exterior angle is equal to the sum of the measures of the two opposite(remote) interior angles of the triangle. Let us recall a few common properties about the angles of a triangle: A triangle has 3 internal angles which always sum up to 180 degrees. It has 6 exterior angles and this theorem gets applied to each of the exterior angles. Note that an exterior angle is supplementary to its adjacent interior angle as they form a linear pair of angles. Exterior angles are defined as the angles formed between the side of the polygon and the extended adjacent side of the polygon.
We can verify the exterior angle theorem with the known properties of a triangle. Consider a Δ ABC.
The three angles a + b + c = 180 (angle sum property of a triangle) ----- Equation 1
c= 180 - (a+b) ----- Equation 2 (rewriting equation 1)
e = 180 - c----- Equation 3 (linear pair of angles)
Substituting the value of c in equation 3, we get
e = 180 - [180 - (a + b)]
e = 180 - 180 + (a + b)
e = a + b
Hence verified.
Proof of Exterior Angle Theorem
Consider a ΔABC. a, b and c are the angles formed. Extend the side BC to D. Now an exterior angle ∠ACD is formed. Draw a line CE parallel to AB. Now x and y are the angles formed, where, ∠ACD = ∠x + ∠y
| Statement | Reason |
|---|---|
| ∠a = ∠x | Pair of alternate angles. (Since BA is parallel to CE and AC is the transversal). |
| ∠b = ∠y | Pair of corresponding angles. (Since BA is parallel to CE and BD is the transversal). |
| ∠a + ∠b = ∠x + ∠y | From the above statements |
| ∠ACD = ∠x + ∠y | From the construction of CE |
| ∠a + ∠b = ∠ACD | From the above statements |
Hence proved that the exterior angle of a triangle is equal to the sum of the two opposite interior angles.
Exterior Angle Inequality Theorem
The exterior angle inequality theorem states that the measure of any exterior angle of a triangle is greater than either of the opposite interior angles. This condition is satisfied by all the six external angles of a triangle.
Related Articles
Check out a few interesting articles related to Exterior Angle Theorem.
- Exterior Angle Formula
- Exterior Angle Theorem Worksheets
- Alternate Exterior Angles
- How to find the measure of each exterior angle of a regular pentagon?
- Properties of Triangle
- Interior and Exterior Angles Worksheets
- Sum of Exterior Angles Formula
Important notes
- The exterior angle theorem states that the measure of an exterior angle is equal to the sum of the measures of the two remote interior angles of the triangle.
- The exterior angle inequality theorem states that the measure of any exterior angle of a triangle is greater than either of the opposite interior angles.
- The exterior angle and the adjacent interior angle are supplementary. All the exterior angles of a triangle sum up to 360º.
Exterior Angle Theorem Examples
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Example 1: Find the values of x and y by using the exterior angle theorem of a triangle.
Solution:
∠x is the exterior angle.
∠x + 92 = 180º (linear pair of angles)
∠x = 180 - 92 = 88º
Applying the exterior angle theorem, we get, ∠y + 41 = 88
∠y = 88 - 41 = 47º
Therefore, the values of x and y are 88º and 47º respectively.
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Example 2: Find ∠BAC and ∠ABC.
Solution:
160º is an exterior angle of the Δ ABC. So, by using the exterior angle theorem, we have, ∠BAC + ∠ABC = 160º
x + 3x = 160º
4x = 160º
x = 40º
Therefore, ∠BAC = x = 40º and ∠ABC = 3xº = 120º
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Example 3: Find ∠ BAC, if ∠CAD = ∠ADC
Solution:
Solving the linear pair at vertex D, we get ∠ADC + ∠ADE = 180º
∠ADC = 180º - 150º = 30º
Using the angle sum property, for Δ ACD,
∠ADC + ∠ACD + ∠CAD = 180º
∠ACD = 180 - ∠CAD -∠ADC
180º - ∠ADC -∠ADC (given ∠CAD= ∠ADC)
180º - 2∠ADC
180º - 2 × 30º
∠ACD = 180º - 60º = 120º
∠ACD is the exterior angle of ∠ABC
Using the exterior angle theorem, for Δ ABC, ∠ACD = ∠ABC + ∠BAC
120º = 60º + ∠BAC
Therefore, ∠BAC = 120º - 60º = 60º.
FAQs on Exterior Angle Theorem
What is the Exterior Angle Theorem?
The exterior angle theorem states that the measure of an exterior angle is equal to the sum of the measures of the two remote interior angles of the triangle. The remote interior angles are also called opposite interior angles.
How do you use the Exterior Angle Theorem?
To use the exterior angle theorem in a triangle we first need to identify the exterior angle and then the associated two remote interior angles of the triangle. A common mistake of considering the adjacent interior angle should be avoided. After identifying the exterior angles and the related interior angles, we can apply the formula to find the missing angles or to establish a relationship between sides and angles in a triangle.
What are Exterior Angles?
An exterior angle of a triangle is formed when any side of a triangle is extended. There are 6 exterior angles of a triangle as each of the 3 sides can be extended on both sides and 6 such exterior angles are formed.
What is the Exterior Angle Inequality Theorem?
The measure of an exterior angle of a triangle is always greater than the measure of either of the opposite interior angles of the triangle.
What is the Exterior Angle Property?
An exterior angle of a triangle is equal to the sum of its two opposite non-adjacent interior angles. The sum of the exterior angle and the adjacent interior angle that is not opposite is equal to 180º.
What is the Exterior Angle Theorem Formula?
The sum of the exterior angle = the sum of two non-adjacent interior opposite angles. An exterior angle of a triangle is equal to the sum of its two opposite non-adjacent interior angles.
Where Should We Use Exterior Angle Theorem?
Exterior angle theorem could be used to determine the measures of the unknown interior and exterior angles of a triangle.
Do All Polygons Exterior Angles Add up to 360?
The exterior angles of a polygon are formed when a side of a polygon is extended. All the exterior angles in all the polygons sum up to 360º.